1. Get a cheatsheet into each player's hands.
You know that godsend player, the one who always has the notecards? Key thing there: the notecards.
You've spoken to the group, and they got upset, but you know they cared enough to get the books in the first place. It's entirely possible that they do just forget, or maybe they're having a difficult time with the rules and don't know how to articulate it. Even if they read cover to cover (and maybe they already have), there's a lot of information to parse and remember, especially for new players. Even veterans forget rules sometimes! Still, fumbling through the book for basic rules is not time-effective. My solution is to create a cheatsheet. This has helped me as both GM and player.
Figure out your most commonly used rules. Type up these rules and print a copy for each player. Try to condense them as much as you can. Strip them to their bare bones so players can find the information they need at a glance. Also, since you'll be distributing the same page to everyone, concentrate on rules that everyone uses. I'll discuss character-specific rules later on.
It's just more practical for you to make the cheatsheet. If they don't have the rules down yet, they probably don't know which ones are the most important. You likely have a better grasp, and you're also the one who wants more efficiency. The best way to ensure that this happens is to do it yourself, and then the whole group will be on the same page.
For character-specific rules, though, it's best if the players do it. This spreads the effort around. Plus, you can use this as an opportunity to help re-familiarize your players with their mechanics. Sit down with them and help them make index cards or type notes from the book. Point out nifty features and answer any questions.
Assure them that it will make their lives way easier, because it will. They probably don't like book-fumbling either. They'll also be more likely to use their character's features if they're always right in front of them in an easily-digestible format. It's hard to achieve your full potential if you don't know where to find it. The key thing is to emphasize how useful it will be for them (in character and out) if they do this, not how much worse things will be if they don't. Make it about them and their potential fun, not your frustration, to avoid further souring.
It's awesome that the Battlemaster Fighter took the initiative, though. If this is something he enjoys doing, you could enlist his help! I'm this player in my own groups, even when I'm not GMing. I love having a cheat sheet, so if I'm making one anyway, why not share? At the least, use his notes as an example (if he's okay with it).
2. Make use of existing resources.
Cheatsheets you make will be the best-tailored to your needs. That doesn't mean you shouldn't use ready-made tools on top of that. People often share their own tools if they think they're especially useful. If you need something specific, search the web to see if it already exists. There's no guarantee that what you find will be useful -- again, you know your own needs best -- but it never hurts to try. If you hit gold, that's time and effort saved.
For instance, here are some form-fillable initiative cards that provide a quick reference of initiative order and PC skills, traits, and actions; here is a larger, longer version of a similar concept. Here's a site that generates spell cards. ENworld has many sheets available. It only took me a few minutes to locate them by searching "5e spell cards," "5e initiative cards," and "5e character sheets," respectively. You'll be more successful if you already know what you need, but you can still try searches like "5e game resources" and "5e player print outs" if you're looking for tools in general.
Give them 'phantom levels'
I ran a game a few years ago, where we had a brand new player (to our table and RPGs in general) join our level 7 party. This was a similar situation to yours, as I had to get them tough enough not to get perma-dead by the first enemy they encountered, but they would have been overwhelmed if I had them make a 7th level character.
What I wish I had done in that situation, and what user T.L.D. has done (for previous editions) was give them the HP, Armor, and Weapons of a level 7 character, with the features of a level 1 character.
My suggestion to you is that they start at level 1, but have the HP that their character would have if they were the same level as your lowest leveled player at the table (or the level of the table if that's how you're doing things).
Then give them any mechanical bonuses commensurate to the party's level that do not increase the complexity of their character. This may include:
increased proficiency bonus
increased HP
better armor (magical if appropriate, limited to +1's and +2's)
better weapons (magical if appropriate, limited to +1's and +2's)
class features that do not induce choice or increase complexity (e.g. Monk's level 6 feature to overcome resistance to non-magical attacks)
Then, after each session (or two), they replace 1 phantom level with a real one. They do not get more health (you already gave that to them), but they add their class features, feats, spells, etc... that a character of their level would have.
This way, you don't have to be afraid to hit them, but they can still learn the character at a normal pace. Obviously, this works best for more martial classes vs casters.
Best Answer
Yes, you should boost their character level
Heterogeneous parties don't work very well in DnD. The game is designed around the whole party being roughly the same level.
When you give the new player a level 1 character, they will be handicapped not just because they lack experience with the game rules but also because their character is less powerful. That will prevent them from contributing to the party in a useful manner and likely end up as a quite frustrating experience for them.
You might want to help them with building their character and recommend them what choices to make at the simulated levelups, because they still lack the game experience to make smart choices in this regard. You might want to recommend them a mechanically simple character build. A front-line tank like a Barbarian or Fighter is likely a good option. Also, don't forget to give them level-appropriate equipment.
The only argument to not boost their level is if you want to give them the experience of working their way up and building their character incrementally while playing. If you want to do that it would be far better to start a new party with everyone in the group rolling a new level 1 character. When the old players don't like that break from their campaign, you could have the stories of both parties take place in the same world at the same time and have their stories intervene with each other. When the new party reached the level of the old party, you could arrange for the two parties to meet. Then every player can decide if they want to continue playing with their old character or with their new one. Your group then forms a new party from their picks with which you then continue the campaign.